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On the modified $J$-equation

Ryosuke Takahashi

TL;DR

This work studies the modified $J$-equation on compact Kähler manifolds with a torus action, establishing that solvability is equivalent to coercivity of the associated modified $J$-functional. It extends the analysis to a generalized equation with parameter $b$, and, in the toric projective setting, proves a Nakai–Moishezon-type criterion characterizing solvability via intersection positivity on toric subvarieties. By combining convexity, subsolution techniques, and a refined modified $J$-flow analysis (including local smoothing and gluing), the authors connect analytic solvability to algebraic stability notions and derive a numerical criterion for the existence of extremal Kähler metrics on smooth projective toric varieties. The results generalize prior work on the $J$-equation and provide a framework for verifying extremal metrics through computable intersection data. Overall, the paper advances the bridge between variational methods for Kähler metrics and algebro-geometric criteria in the presence of symmetries.

Abstract

In this paper, we study the modified $J$-equation introduced by Li-Shi. We show that the solvability of the modified $J$-equation is equivalent to the coercivity of the modified $J$-functional on compact Kähler manifolds. For smooth projective toric varieties we establish a Nakai-Moishezon type criterion for the existence of solutions, extending the results of Collins-Székelyhidi. As a possible application, by combining our result with Delcroix-Jubert, we further discuss a numerical sufficient condition for the existence of extremal Kähler metrics on smooth projective toric varieties.

On the modified $J$-equation

TL;DR

This work studies the modified -equation on compact Kähler manifolds with a torus action, establishing that solvability is equivalent to coercivity of the associated modified -functional. It extends the analysis to a generalized equation with parameter , and, in the toric projective setting, proves a Nakai–Moishezon-type criterion characterizing solvability via intersection positivity on toric subvarieties. By combining convexity, subsolution techniques, and a refined modified -flow analysis (including local smoothing and gluing), the authors connect analytic solvability to algebraic stability notions and derive a numerical criterion for the existence of extremal Kähler metrics on smooth projective toric varieties. The results generalize prior work on the -equation and provide a framework for verifying extremal metrics through computable intersection data. Overall, the paper advances the bridge between variational methods for Kähler metrics and algebro-geometric criteria in the presence of symmetries.

Abstract

In this paper, we study the modified -equation introduced by Li-Shi. We show that the solvability of the modified -equation is equivalent to the coercivity of the modified -functional on compact Kähler manifolds. For smooth projective toric varieties we establish a Nakai-Moishezon type criterion for the existence of solutions, extending the results of Collins-Székelyhidi. As a possible application, by combining our result with Delcroix-Jubert, we further discuss a numerical sufficient condition for the existence of extremal Kähler metrics on smooth projective toric varieties.
Paper Structure (21 sections, 36 theorems, 274 equations)

This paper contains 21 sections, 36 theorems, 274 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convexity properties
  4. The $b=-\varepsilon<0$ case
  5. The $b \geqslant 0$ case
  6. Hamiltonians
  7. Subsolutions
  8. Functionals on the space of Kähler potentials
  9. The modified $J$-flow
  10. Long-time existence for $b=-\varepsilon<0$
  11. Convergence for $b \geqslant 0$
  12. Coercivity of the modified $J$-functional
  13. Coercivity implies existence
  14. Local smoothing
  15. Gluing argument
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $X$ be a compact complex manifold, and $T$, $v$, $\omega$, $\widehat{\chi}$ as above. Then the generalized equation generalized modified J-equation with $b \geqslant 0$ admits a solution if and only if there exists $\underline{\phi} \in \mathcal{H}(X,\widehat{\chi})^T$ such that

Theorems & Definitions (71)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Lemma 2.1
  • proof
  • ...and 61 more